The answer is yes it is possible to claim Unemployment Insurance if you move between States. All States are involved in the Federal Interstate Benefit Payment Plan. There are strict rules to be followed when applying and if these are not followed the claim may become invalid.
Firstly there is the need to find out how to apply. In some cases it may be possible to do it online or by phone. The office nearest to were the new residence is will be able to give all the details. If a visit is necessary there may be the need to make an appointment. Always ask about the documents needed as it could turn into a wasted trip and a longer period without any money.
It is more than likely that all the forms will be available when the visit is made so as long as all the paperwork from the former state is correct the process should run smoothly. The forms will be sent to the former State as they will be the ones providing the payment.
An agent will take the completed paperwork and then the information as to how the claims can be named will be passed on. Sometimes the claim can be put in weekly over the phone. It may be best to expect to wait a couple of weeks before asking for duplicate forms as it could take a while for the paperwork to be sorted.
File weekly claims with the liable state after serving the "waiting week." After the waiting week, collect unemployment benefits. Continue to seek work and document all work search efforts. If required to provide documentation by the liable state of work search efforts, do so in a timely and thorough manner to the agent state.
If work has not been found once the benefits run out it will be up to the new state to decide if it is possible to ask the former state to extend the period that can be claimed.